Size-dependent energy in crystal plasticity and continuum dislocation models

On a mixed energetic–dissipative constitutive law for non-proportional loading, with focus on small-scale plasticity

We analyse the mixed energetic–dissipative potential (MP) recently proposed by our group to predict, within higher-order strain gradient plasticity (SGP), reliable size-dependent responses under general loading histories. Such an MP follows former proposals by Chaboche, Ohno and co-workers for nonlinear kinematic hardening in the context of size-independent metal plasticity. The MP is given by M quadratic addends that each transitions, at a different threshold value, into a linear dissipative contribution. Hence, the MP involves 2 M positive material parameters, given by the M threshold values and the M moduli weighing each quadratic recoverable term. We analytically demonstrate that, under proportional loading, the MP limit for M  → ∞ converges to a less-than-quadratic potential with well-defined properties. This result is of crucial importance for identifying the material parameters of any model adopting the MP. Moreover, our analysis provides a formula for the characterization of the energetic and dissipative parts of any possible MP limit, showing that, regarding the capability to describe the effect of diminishing size within SGP, the MP can be selected such that its contribution to the strengthening (i.e. an increase in yield point) is mostly dissipative, whereas its contribution to the increase in strain hardening is mostly recoverable.

A potential for higher-order phenomenological strain gradient plasticity to predict reliable response under non-proportional loading

We propose a plastic potential for higher-order (HO) phenomenological strain gradient plasticity (SGP), predicting reliable size-dependent response for general loading histories. By constructing the free energy density as a sum of quadratic plastic strain gradient contributions that each transitions into linear terms at different threshold values, we show that we can predict the expected micron-scale behaviour, including increase of strain hardening and strengthening-like behaviour with diminishing size. Furthermore, the anomalous behaviour predicted by most HO theories under non-proportional loading is avoided. Though we demonstrate our findings on the basis of Gurtin (Gurtin 2004 J. Mech. Phys. Solids 52, 2545-2568, doi:10.1016/j.jmps.2003.11.002) distortion gradient plasticity, adopting Nye's dislocation density tensor as primal HO variable, we expect our results to hold qualitatively for any HO SGP theory, including crystal plasticity.